Elasticity in Apery sets

TL;DR

This paper investigates the properties of Apery sets in numerical semigroups, focusing on the non-uniqueness of element factorizations and introducing new invariants to measure this phenomenon.

Contribution

It introduces new invariants to quantify non-unique factorizations within Apery sets of numerical semigroups, advancing understanding of their algebraic structure.

Findings

Identification of elements with non-unique factorizations in Apery sets

Development of new invariants to measure factorization non-uniqueness

Insights into the structure of numerical semigroups through these invariants

Abstract

A numerical semigroup $S$ is an additive subsemigroup of the non-negative integers, containing zero, with finite complement. Its multiplicity $m$ is its smallest nonzero element. The Apery set of $S$ is the set $\text{Ap}(S) = \{n \in S : n-m \notin S\}$. Fixing a numerical semigroup, we ask how many elements of its Apery set have nonunique factorization, and define several new invariants.

Equations

$$\mathrm{Ap}(S)=\{x\in S:x-m(S)\notin S\},$$

$$\mathrm{Ap}(S)=\{0,a,2a,\dots ,(m-1)a\}.$$

$$\mathrm{Ap}(S)=\{0,a_1,a_2,2a_1,3a_1=2a_2\},$$

$$R(T)=\{\rho (n):n\in T\}.$$

$$\mathrm{Ap}(S)=\{0,pb,2pb,\dots ,(a-1)pb,pa,2pa,\dots ,(b-1)pa,pab\},$$

$$\mathrm{Ap}(S)=\{0,2p,3p,\dots ,(5n-1)p,(5n+1)p\}.$$

$$MAE(S)=\frac{1}{|\mathrm{Ap}(S)|}\sum _{n\in \mathrm{Ap}(S)}\rho (n).$$

$$Ap(S)=\{0,2p,4p,\dots ,(q-1)p,qp,(q+1)p,\dots ,\frac{1}{2}(9q+17)p\},$$

$$T=\{(2q+2i)p:0\le i<q\}\subset \mathrm{Ap}(S).$$

$$\rho ((2q+2i)p)=\rho ((q+i)a_1)=\rho (2a_2+i{a}_1)=\frac{q+i}{2+i}\ge \frac{q}{2+i},$$

$$\begin{array}[]{r@{}c@{}l}MAE(S)&{}={}&\displaystyle\frac{1}{m}\sum_{n\in\operatorname{Ap}(S)}\rho(n)\geq\frac{3q+8}{m}+\frac{1}{m}\sum_{n\in T}\rho(n)\geq\frac{3q+8}{m}+\frac{q}{m}\sum_{i=0}^{q-1}\frac{1}{2+i}\\ \end{array}$$

$$\underset{g\to \infty }{\lim }\frac{h_{m,g}}{n_{m,g}}=1.$$

$$0<\underset{g\to \infty }{\lim }\frac{h_g}{n_g}<1.$$

$$\underset{g\to \infty }{\lim }\frac{f_{g+1}}{n_g}>0.$$

$$a_i=\{\begin{array}{ll}2m+i& \text{if }i\in T;\\ m+i& \text{if }i\notin T.\end{array}$$

$$h_g=\sum _{m=2}^{g+1}{h}_{m,g}\ge \sum _{m=2}^{g+1}\left(\genfrac{}{}{0ex}{}{m-1}{g-(m-1)}\right)=f_{g+1}.$$

$$a_3=3a_1=a_4+a_{m-1}$$

Full text

1. Introduction

Every child’s first semigroup is the natural numbers, and their first factorization theorem is the Fundamental Theorem of Arithmetic, which gives unique factorization as a product of primes. The other operation, addition, is not addressed. Much attention has been given to factorization in various semigroups; for a general introduction, see [10]. Often, the operation is multiplication [4, 7, 12], but addition is worth studying as well [19]; it will be our operation here.

A numerical semigroup $S$ is a subset of $\mathbb{Z}_{\geq 0}$ with finite complement that is closed under $+$ and contains [math]. Numerical semigroups have been the subject of considerable recent study [8, 11, 14, 15, 16, 17]. Many applications are known, such as in coding theory [6]. For a general introduction to numerical semigroups, see [3] or [18].

The atoms of a numerical semigroup $S$ are the nonzero elements that cannot be expressed as the combination of two nonzero elements. The set $\mathcal{A}(S)$ of atoms of $S$ is finite; we call $e(S)=|\mathcal{A}(S)|$ the embedding dimension of $S$. We write $\left\langle a_{1},a_{2},\ldots,a_{k}\right\rangle$, with $a_{i}$ listed in ascending order, to denote the numerical semigroup with atoms $a_{1},\ldots,a_{k}$. The smallest atom $a_{1}$ is also the smallest nonzero element of $S$; we call it $m(S)$, the multiplicity of $S$.

An important tool for the study of numerical semigroups, from [2], is the Apéry set

[TABLE]

which contains the smallest element of $S$ in each congruence class modulo $m(S)$. It is easy to show that $|\operatorname{Ap}(S)|=m(S)$ and $\mathcal{A}(S)\setminus\{m(S)\}\subseteq\operatorname{Ap}(S)$. If we want to express elements of $S$ as a free combination of atoms, $\mathcal{A}(S)$ is what we study. However, if we want to use as many copies of $m(S)$ as possible and other atoms as little as possible, we look to $\operatorname{Ap}(S)$ instead.

We study properties about factorization into atoms. The most famous factorization invariant is elasticity. Given a semigroup $S$ and some $x\in S$, we write $x$ as the combination of atoms in every possible way. The elasticity of $x$, denoted $\rho(x)$, is the the largest number of atoms that can be used, divided by the smallest number. Clearly $\rho(x)\geq 1$; if equality holds, we say $x$ is half-factorial. Conventionally we say that the unit [math] is half-factorial.

In this note, we consider the elasticity function restricted to elements of $\operatorname{Ap}(S)$. We write $\rho(\operatorname{Ap}(S))$ for the maximum elasticity over the elements of $\operatorname{Ap}(S)$; if $\rho(\operatorname{Ap}(S))=1$, we say that $S$ is Apéry half factorial, or AHF.

We can visualize the factorization structure of $\operatorname{Ap}(S)$ using a partially ordered set $(\operatorname{Ap}(S),\preceq)$ with $x\preceq y$ whenever $y-x\in S$, called the Apéry poset of $S$. The Hasse diagrams of two Apéry posets are depicted in Figure 1. The atoms of the Apéry poset (i.e., the elements directly above the unique minimal element [math]) are precisely the elements of $\mathcal{A}(S)$ apart from $m(S)$, and an edge connects $x$ up to $y$ in the Hasse diagram exactly when $y-x\in\mathcal{A}(S)$. This leads to the following interesting observation.

Theorem 1.1**.**

A numerical semigroup $S$ has graded Apéry poset if and only if $S$ is AHF.

Proof.

By the above discussion, each length $\ell$ chain (set of mutually comparable elements) from [math] to an element $n\in S$ corresponds to an ordered factorization of $n$ with length $\ell$. As such, two different chain lengths are present if and only if $n$ is not half-factorial. ∎

2. Apéry elasticities

We begin by observing that if $e(S)=2$, then we can write $S=\left\langle m,a\right\rangle$ and

[TABLE]

Each element of $\operatorname{Ap}(S)$ is then not only half-factorial, but has unique factorization. On the other hand, if $S$ has maximal embedding dimension (i.e., $e(S)=m(S)$), then $S=\left\langle m,a_{1},a_{2},\ldots,a_{m-1}\right\rangle$ and $\operatorname{Ap}(S)=\{0,a_{1},a_{2},\ldots,a_{m-1}\}$. Again each element of $\operatorname{Ap}(S)$ has unique factorization. These observations are extended slightly as follows.

Theorem 2.1**.**

Let $S$ be a numerical semigroup with $e(S)=2$ or $e(S)\geq m(S)-1$. Then $S$ is Apéry half-factorial.

Proof.

If $e(S)=m(S)-1$, then the only element of $\operatorname{Ap}(S)$ that is not an atom only has length 2 factorizations. ∎

Theorem 2.1 can’t be extended in general to smaller embedding dimension than $m(S)-1$. Consider $S=\left\langle 5,6,9\right\rangle$, where $m=5$, $a_{1}=6,a_{2}=9$. Now

[TABLE]

so $\rho(3a_{1})=\frac{3}{2}$.

Given a subset $T\subset S$, define the set of elasticities of $T$ as

[TABLE]

This invariant has been studied for numerical semigroups in [5], wherein $R(S)$ is characterized for all but finitely many elasticities coming from “small” elements of $S$. As such, $R(\operatorname{Ap}(S))$ is a natural starting place for studying the remainder of $R(S)$.

It is easy to see that if $e(S)=m(S)-2$, either $S$ is AHF or $R(\operatorname{Ap}(S))=\{1,\frac{3}{2}\}$. Determining $R(\operatorname{Ap}(S))$ for other near-maximal embedding dimensions remains open.

We now present a family of semigroups in Theorem 2.2 which demonstrate several extremal behaviors, as discussed thereafter.

Theorem 2.2**.**

Fix $a>b\geq 1$ with $\gcd(a,b)=1$. There is a numerical semigroup $S$ with (i) $R(\operatorname{Ap}(S))=\{1,\tfrac{a}{b}\}$ and (ii) only one element of $\operatorname{Ap}(S)$ has elasticity $\tfrac{a}{b}$.

Proof.

Fix a prime $p\nmid(a+b)$ with $a+b<pb$, and let $S=\left\langle a+b,pa,pb\right\rangle$. We have

[TABLE]

wherein each element has unique factorization except $pab$, which has elasticity $\tfrac{a}{b}$. ∎

One natural question to ask is: which subsets of $\mathbb{Q}_{\geq 1}$ can occur as $R(\operatorname{Ap}(S))$ for some numerical semigroup $S$? Certainly we must have $1\in R(\operatorname{Ap}(S))$, and the sole singleton subset, $\{1\}$, is achieved for all Apéry half-factorial $S$. All subsets of size two are realizable by Corollary 2.3. Larger subsets of $\mathbb{Q}_{\geq 1}$ remain unresolved.

Corollary 2.3**.**

Given $r\in\mathbb{Q}_{>1}$, some numerical semigroup $S$ has $R(\operatorname{Ap}(S))=\{1,r\}$.

Proof.

Write $r=\frac{a}{b}$ in reduced form, and apply Theorem 2.2. ∎

Since $\operatorname{Ap}(S)$ is a finite set, we can consider the full distribution of elasticity over its elements, and not just its maximum $\rho(\operatorname{Ap}(S))$. We call the Apéry half-factorial fraction, or AHFF, the ratio of the number of half-factorial elements of $\operatorname{Ap}(S)$, to $|\operatorname{Ap}(S)|$. If $S$ is AHF, then its AHFF is $1$.

Theorem 2.2 produced a single non-half-factorial element of $\operatorname{Ap}(S)$; hence $S$ had AHFF close to $1$. Certainly the AHFF cannot be zero, as each element of $\mathcal{A}(S)$ is half-factorial. One wonders how small the AHFF can be. Theorem 2.4, illustrated in Figure 1(a), displays the smallest possible AHFF while maintaining $e(S)=3$.

Theorem 2.4**.**

The fraction of Apéry set elements of a numerical semigroup that are half-factorial can be arbitrarily close to [math].

Proof.

Let $p,n\in\mathbb{Z}_{\geq 1}$ with $p$ prime, $p\neq 5$, and $2p>5n>5$. Set $m=5n$, $a_{1}=2p$, $a_{2}=3p$, and take $S=\left\langle m,a_{1},a_{2}\right\rangle$. We have

[TABLE]

Since $6p=3a_{1}=2a_{2}$, only [math], $2p$, $3p$, $4p$, $5p$ and $7p$ are half-factorial in $\operatorname{Ap}(S)$. As such, the AHFF of $S$ is $\frac{6}{5n}$. ∎

With the generality of the family in Theorem 2.2, one might wonder if any $S$ with $e(S)=3$ can be AHF. One such family is provided in Theorem 2.5, an example of which is illustrated in Figure 1(b).

Theorem 2.5**.**

For each $n\in\mathbb{Z}_{\geq 2}$, the semigroup $S=\left\langle n^{2},n^{2}+n,2n^{2}+1\right\rangle$ is AHF.

Proof.

$\operatorname{Ap}(S)=\{a(n^{2}+n)+b(2n^{2}+1):0\leq a,b\leq n-1\}$. ∎

Theorem 2.5 also demonstrates that the width of the Apéry poset, which is always bounded below by $e(S)$, can be larger.

Corollary 2.6**.**

The width of an Apéry poset can be arbitrarily large, even for $e(S)=3$.

3. Mean Apéry elasticity

Motivated in part by recent investigations into “average” factorization lengths in numerical semigroups [9], we next consider the mean Apéry elasticity, i.e.,

[TABLE]

If $S$ is half-factorial, of course $MAE(S)=1$. The family from Corollary 2.3 has mean Apéry elasticity $1+\frac{1}{b}-\frac{2}{a+b}$. Theorem 3.1 will show that mean Apéry elasticity may be arbitrarily large, though one may still wonder which elements of $\mathbb{Q}_{\geq 1}$ occur as $MAE(S)$ for some numerical semigroup $S$.

Theorem 3.1**.**

The values of $MAE(S)$, with $e(S)=3$, can be arbitrarily large.

Proof.

Let $p,q$ be odd primes with $p>2q+4$. Set $m=4q+8,a_{1}=2p,a_{2}=qp$, and take $S=\left\langle m,a_{1},a_{2}\right\rangle$. We have

[TABLE]

where all multiples of $p$ are present after $qp$ except $(4q+8)p$. Now, consider the set

[TABLE]

We calculate elasticity of the elements of $T$ as

[TABLE]

and consequently

[TABLE]

grows arbitrarily large as $q\to\infty$. ∎

4. Asymptotic distributions

Given a numerical semigroup $S$, denote by $g(S)=|\mathbb{Z}_{\geq 0}\setminus S|$ the genus of $S$. Let $n_{g}$ denote the number of numerical semigroups with genus $g$, and let $n_{m,g}$ denote the number of numerical semigroups with multiplicity $m$ and genus $g$. For example, letting $f_{g}$ denote the $g$’th Fibonacci number, it was recently proven that $n_{g}/f_{g}$ approaches a constant as $g\to\infty$ [20], although it is still open whether $n_{g+1}\geq n_{g}$ for every $g\geq 0$. On the other hand, for fixed $m$, the ratio $n_{m,g}/g^{m-1}$ approaches a constant as $g\to\infty$.

There has been a recent push to understand the distribution of numerical semigroups with a given genus across different special families. For example, if $M_{g}$ and $M_{m,g}$ denote, respectively, the number of maximal embedding dimension numerical semigroups with genus $g$ and the number with both multiplicity $m$ and genus $g$, then $M_{g}/n_{g}\to 0$ as $g\to\infty$, while $M_{m,g}/n_{m,g}\to 1$ as $g\to\infty$; see [1, 13].

Continuing in this vein, let $h_{g}$ denote the number of AHF numerical semigroups with genus $g$, and let $h_{m,g}$ denote the number of AHF numerical semigroups with multiplicity $m$ and genus $g$. Theorems 4.1 and 4.2 below demonstrate that AHF numerical semigroups form a much larger class than those with maximum embedding dimension.

Theorem 4.1**.**

For each fixed $m\geq 2$, we have

[TABLE]

Proof.

Apply [1, Corollary 1]. ∎

Identifying the precise value of the limit below will likely be challenging, considering the long and technical nature of the proof of [20, Theorem 1]. Out of the 1179593 numerical semigroups with genus at most 25, we find 1032971 (about 88%) are AHF.

Theorem 4.2**.**

We have

[TABLE]

Proof.

Let $f_{n}$ denote the $n$’th Fibonacci number. By [20, Theorem 1], we have

[TABLE]

As such, for the first inequality, it suffices to show that $f_{g+1}\leq h_{g}$. Fix a multiplicity $m\leq g+1$. For each subset $T\subset\{1,\ldots,m-1\}$, consider the numerical semigroup $S$ with Apéry set given by $\operatorname{Ap}(S)=\{0,a_{1},\ldots,a_{m-1}\}$, where

[TABLE]

It is clear $S$ has multiplicity $m$ and genus $m+|T|$, and is AHF. As such,

[TABLE]

For the other inequality, we use a similar construction, where we first let $a_{1}=m+1$, $a_{2}=2m+2$, $a_{3}=3m+3$, $a_{4}=m+4$, and $a_{m-1}=m+(m-1)$, and then choose the remaining $a_{i}$ as above. In each resulting semigroup,

[TABLE]

is not half-factorial, and by similar reasoning to above, this family of semigroups also comprises a positive asymptotic proportion of those with genus $g$. ∎